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© 2002 Prentice-Hall, Inc.Chap 4-1 Statistics for Managers Using Microsoft Excel (3 rd Edition) Chapter 4 Basic Probability and Discrete Probability Distributions

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© 2002 Prentice-Hall, Inc. Chap 4-2 Chapter Topics Basic probability concepts Sample spaces and events, simple probability, joint probability Conditional probability Statistical independence, marginal probability Bayes’s Theorem

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© 2002 Prentice-Hall, Inc. Chap 4-3 Chapter Topics The probability of a discrete random variable Covariance and its applications in finance Binomial distribution Poisson distribution Hypergeometric distribution (continued)

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© 2002 Prentice-Hall, Inc. Chap 4-4 Sample Spaces Collection of all possible outcomes e.g.: All six faces of a die: e.g.: All 52 cards in a deck:

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© 2002 Prentice-Hall, Inc. Chap 4-5 Events Simple event Outcome from a sample space with one characteristic e.g.: A red card from a deck of cards Joint event Involves two outcomes simultaneously e.g.: An ace that is also red from a deck of cards

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© 2002 Prentice-Hall, Inc. Chap 4-6 Visualizing Events Contingency Tables Tree Diagrams Red 2 24 26 Black 2 24 26 Total 4 48 52 Ace Not Ace Total Full Deck of Cards Red Cards Black Cards Not an Ace Ace Not an Ace

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© 2002 Prentice-Hall, Inc. Chap 4-7 Simple Events The Event of a Triangle 5 There are 5 triangles in this collection of 18 objects

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© 2002 Prentice-Hall, Inc. Chap 4-8 The event of a triangle AND blue in color Joint Events Two triangles that are blue

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© 2002 Prentice-Hall, Inc. Chap 4-9 Special Events Impossible event e.g.: Club & diamond on one card draw Complement of event For event A, all events not in A Denoted as A’ e.g.: A: queen of diamonds A’: all cards in a deck that are not queen of diamonds Null Event

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© 2002 Prentice-Hall, Inc. Chap 4-10 Special Events Mutually exclusive events Two events cannot occur together e.g.: -- A: queen of diamonds; B: queen of clubs Events A and B are mutually exclusive Collectively exhaustive events One of the events must occur The set of events covers the whole sample space e.g.: -- A: all the aces; B: all the black cards; C: all the diamonds; D: all the hearts Events A, B, C and D are collectively exhaustive Events B, C and D are also collectively exhaustive (continued)

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© 2002 Prentice-Hall, Inc. Chap 4-11 Contingency Table A Deck of 52 Cards Ace Not an Ace Total Red Black Total 224 2 26 44852 Sample Space Red Ace

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© 2002 Prentice-Hall, Inc. Chap 4-12 Full Deck of Cards Tree Diagram Event Possibilities Red Cards Black Cards Ace Not an Ace Ace Not an Ace

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© 2002 Prentice-Hall, Inc. Chap 4-13 Probability Probability is the numerical measure of the likelihood that an event will occur Value is between 0 and 1 Sum of the probabilities of all mutually exclusive and collective exhaustive events is 1 Certain Impossible.5 1 0

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© 2002 Prentice-Hall, Inc. Chap 4-14 (There are 2 ways to get one 6 and the other 4) e.g. P ( ) = 2/36 Computing Probabilities The probability of an event E: Each of the outcomes in the sample space is equally likely to occur

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© 2002 Prentice-Hall, Inc. Chap 4-15 Computing Joint Probability The probability of a joint event, A and B:

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© 2002 Prentice-Hall, Inc. Chap 4-16 P(A 1 and B 2 )P(A 1 ) Total Event Joint Probability Using Contingency Table P(A 2 and B 1 ) P(A 1 and B 1 ) Event Total 1 Joint Probability Marginal (Simple) Probability A1A1 A2A2 B1B1 B2B2 P(B 1 ) P(B 2 ) P(A 2 and B 2 ) P(A 2 )

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© 2002 Prentice-Hall, Inc. Chap 4-17 Computing Compound Probability Probability of a compound event, A or B:

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© 2002 Prentice-Hall, Inc. Chap 4-18 P(A 1 ) P(B 2 ) P(A 1 and B 1 ) Compound Probability (Addition Rule) P(A 1 or B 1 ) = P(A 1 ) + P(B 1 ) - P(A 1 and B 1 ) P(A 1 and B 2 ) Total Event P(A 2 and B 1 ) Event Total 1 A1A1 A2A2 B1B1 B2B2 P(B 1 ) P(A 2 and B 2 ) P(A 2 ) For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

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© 2002 Prentice-Hall, Inc. Chap 4-19 Computing Conditional Probability The probability of event A given that event B has occurred:

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© 2002 Prentice-Hall, Inc. Chap 4-20 Conditional Probability Using Contingency Table Black Color Type Red Total Ace 224 Non-Ace 24 48 Total 26 52 Revised Sample Space

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© 2002 Prentice-Hall, Inc. Chap 4-21 Conditional Probability and Statistical Independence Conditional probability: Multiplication rule:

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© 2002 Prentice-Hall, Inc. Chap 4-22 Conditional Probability and Statistical Independence Events A and B are independent if Events A and B are independent when the probability of one event, A, is not affected by another event, B (continued)

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© 2002 Prentice-Hall, Inc. Chap 4-23 Bayes’s Theorem Adding up the parts of A in all the B’s Same Event

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© 2002 Prentice-Hall, Inc. Chap 4-24 Bayes’s Theorem Using Contingency Table Fifty percent of borrowers repaid their loans. Out of those who repaid, 40% had a college degree. Ten percent of those who defaulted had a college degree. What is the probability that a randomly selected borrower who has a college degree will repay the loan?

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© 2002 Prentice-Hall, Inc. Chap 4-25 Bayes’s Theorem Using Contingency Table (continued) Repay College 1.0.5.2.3.05.45.25.75 Total

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© 2002 Prentice-Hall, Inc. Chap 4-26 Random Variable Outcomes of an experiment expressed numerically e.g.: Toss a die twice; count the number of times the number 4 appears (0, 1 or 2 times)

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© 2002 Prentice-Hall, Inc. Chap 4-27 Discrete Random Variable Discrete random variable Obtained by counting (1, 2, 3, etc.) Usually a finite number of different values e.g.: Toss a coin five times; count the number of tails (0, 1, 2, 3, 4, or 5 times)

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© 2002 Prentice-Hall, Inc. Chap 4-28 Probability Distribution Values Probability 01/4 =.25 12/4 =.50 21/4 =.25 Discrete Probability Distribution Example T T TT Event: Toss two coinsCount the number of tails

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© 2002 Prentice-Hall, Inc. Chap 4-29 Discrete Probability Distribution List of all possible [X j, p(X j ) ] pairs X j = value of random variable P(X j ) = probability associated with value Mutually exclusive (nothing in common) Collectively exhaustive (nothing left out)

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© 2002 Prentice-Hall, Inc. Chap 4-30 Summary Measures Expected value (the mean) Weighted average of the probability distribution e.g.: Toss 2 coins, count the number of tails, compute expected value

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© 2002 Prentice-Hall, Inc. Chap 4-31 Summary Measures Variance Weight average squared deviation about the mean e.g. Toss two coins, count number of tails, compute variance (continued)

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© 2002 Prentice-Hall, Inc. Chap 4-32 Covariance and its Application

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© 2002 Prentice-Hall, Inc. Chap 4-33 Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession-$100 -$200.5 Stable Economy+ 100 + 50.3 Expanding Economy + 250 + 350 Investment

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© 2002 Prentice-Hall, Inc. Chap 4-34 Computing the Variance for Investment Returns P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession-$100 -$200.5 Stable Economy+ 100 + 50.3 Expanding Economy + 250 + 350 Investment

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© 2002 Prentice-Hall, Inc. Chap 4-35 Computing the Covariance for Investment Returns P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession-$100 -$200.5 Stable Economy+ 100 + 50.3 Expanding Economy + 250 + 350 Investment The Covariance of 23,000 indicates that the two investments are positively related and will vary together in the same direction.

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© 2002 Prentice-Hall, Inc. Chap 4-36 Important Discrete Probability Distributions Discrete Probability Distributions BinomialHypergeometricPoisson

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© 2002 Prentice-Hall, Inc. Chap 4-37 Binomial Probability Distribution ‘n’ identical trials e.g.: 15 tosses of a coin; ten light bulbs taken from a warehouse Two mutually exclusive outcomes on each trials e.g.: Head or tail in each toss of a coin; defective or not defective light bulb Trials are independent The outcome of one trial does not affect the outcome of the other

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© 2002 Prentice-Hall, Inc. Chap 4-38 Binomial Probability Distribution Constant probability for each trial e.g.: Probability of getting a tail is the same each time we toss the coin Two sampling methods Infinite population without replacement Finite population with replacement (continued)

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© 2002 Prentice-Hall, Inc. Chap 4-39 Binomial Probability Distribution Function Tails in 2 Tosses of Coin X P(X) 0 1/4 =.25 1 2/4 =.50 2 1/4 =.25

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© 2002 Prentice-Hall, Inc. Chap 4-40 Binomial Distribution Characteristics Mean E.g. Variance and Standard Deviation E.g. n = 5 p = 0.1 0.2.4.6 012345 X P(X)

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© 2002 Prentice-Hall, Inc. Chap 4-41 Binomial Distribution in PHStat PHStat | probability & prob. Distributions | binomial Example in excel spreadsheet

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© 2002 Prentice-Hall, Inc. Chap 4-42 Poisson Distribution Poisson Process: Discrete events in an “interval” The probability of One Success in an interval is stable The probability of More than One Success in this interval is 0 The probability of success is independent from interval to interval e.g.: number of customers arriving in 15 minutes e.g.: number of defects per case of light bulbs PXx x x (| ! e -

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© 2002 Prentice-Hall, Inc. Chap 4-43 Poisson Probability Distribution Function e.g.: Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6.

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© 2002 Prentice-Hall, Inc. Chap 4-44 Poisson Distribution in PHStat PHStat | probability & prob. Distributions | Poisson Example in excel spreadsheet

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© 2002 Prentice-Hall, Inc. Chap 4-45 Poisson Distribution Characteristics Mean Standard Deviation and Variance = 0.5 = 6 0.2.4.6 012345 X P(X) 0.2.4.6 0246810 X P(X)

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© 2002 Prentice-Hall, Inc. Chap 4-46 Hypergeometric Distribution “n” trials in a sample taken from a finite population of size N Sample taken without replacement Trials are dependent Concerned with finding the probability of “X” successes in the sample where there are “A” successes in the population

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© 2002 Prentice-Hall, Inc. Chap 4-47 Hypergeometric Distribution Function E.g. 3 Light bulbs were selected from 10. Of the 10 there were 4 defective. What is the probability that 2 of the 3 selected are defective?

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© 2002 Prentice-Hall, Inc. Chap 4-48 Hypergeometric Distribution Characteristics Mean Variance and Standard Deviation Finite Population Correction Factor

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© 2002 Prentice-Hall, Inc. Chap 4-49 Hypergeometric Distribution in PHStat PHStat | probability & prob. Distributions | Hypergeometric … Example in excel spreadsheet

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© 2002 Prentice-Hall, Inc. Chap 4-50 Chapter Summary Discussed basic probability concepts Sample spaces and events, simple probability, and joint probability Defined conditional probability Statistical independence, marginal probability Discussed Bayes’s theorem

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© 2002 Prentice-Hall, Inc. Chap 4-51 Chapter Summary Addressed the probability of a discrete random variable Defined covariance and discussed its application in finance Discussed binomial distribution Addressed Poisson distribution Discussed hypergeometric distribution (continued)

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